INMO 2021

Suppose $r \geq 2$ is an integer, and let $m_{1}, n_{1}, m_{2}, n_{2}, \cdots, m_{r}, n_{r}$ be $2 r$ integers such that

$$\left|m_{i} n_{j}-m_{j} n_{i}\right|=1$$

for any two integers $i$ and $j$ satisfying $1 \leq i<j \leq r$. Determine the maximum possible value of $r$.

Find all pairs of integers $(a, b)$ so that each of the two cubic polynomials

$$x^{3}+a x+b \text { and } x^{3}+b x+a$$

has all the roots to be integers.

Betal marks 2021 points on the plane such that no three are collinear, and draws all possible line segments joining these. He then chooses any 1011 of these line segments, and marks their midpoints. Finally, he chooses a line segment whose midpoint is not marked yet, and challenges Vikram to construct its midpoint using only a straightedge. Can Vikram always complete this challenge?

Note: A straightedge is an infinitely long ruler without markings, which can only be used to draw the line joining any two given distinct points.

Let $\mathbb{R}[x]$ be the set of all polynomials with real coefficients, and let $\operatorname{deg} P$ denote the degree of a nonzero polynomial $P$. Find all functions $f: \mathbb{R}[x] \rightarrow \mathbb{R}[x]$ satisfying the following conditions:

• $f$ maps the zero polynomial to itself,
• for any non-zero polynomial $P \in \mathbb{R}[x]$, $\operatorname{deg} f(P) \leq 1+\operatorname{deg} P$, and
• for any two polynomials $P, Q \in \mathbb{R}[x]$, the polynomials $P-f(Q)$ and $Q-f(P)$ have the same set of real roots.